Death.
No, wait.
Taxes.
The suckitude of my life.
No, wait. That’s my least favorite constant.
Pick your favorite well-ordering on R and choose the least undefinable number in [0, 1]. Couldn’t be simpler.
“Cook’s Constant”
The number that when added to, or subtracted from, the answer you got, gives you the answer you should have got. 
I’m going to put in a vote for the fine structure constant, which is discussed here: http://en.wikipedia.org/wiki/Fine_structure_constant
That 1/137 just gives me goosebumps!
Favorite is ε[sub]0[/sub], the permittivity of free space. It’s just really useful.
I absolutely hate Planck’s (and Dirac’s) constant. They remind me of extremely bad times in my life.
Another vote for pi. Picturing a polygon (with its perimeter and the altitude of one of its sides drawn in a thick dark line) with an ever increasing number of sides eventually morphing into a perfect circle, well, it makes my heart go all fluttery and my eyes get that far-away dreamy look.
“John Constantine: Hellblazer.”
My favorite is the Hubble constant, because it can’t yet be measured accurately, and may not actually be constant. Premature nomenclatural assumptions tickle me in that special place. 
Twenty-nine posts and no one’s mentioned my choice, the speed of light in vacuum? I don’t know whether to be pleased or shocked.
c = 3.00 x 10^8 m/s, so beautiful and easy to make estimations with.
i = (-1)^(1/2)
In eleventh grade pre-calc, our first assignment of the year was to write about our feelings on math. I wrote a fairie tale based around imaginary numbers. They bring me great joy.
That is my favorite, but it really isn’t 1/137 any more than pi is 22/7.
I like *i * a whole lot too, and especially -i ::brain explodes
It looks lemon yellow with a hint of lime in my mind.